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8. Domain Transforms
 
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The Laplace and Fourier transforms that serve to transform to the frequency domain can be used to get generalizations of the derivative valid for functions that allow such transformations. The Laplace transform is defined by

(8.1)

while its inverse transform is

(8.2)

where a is chosen so that it is greather than the real part of any of the singularities of f(x). An important property of the Laplace transform is related to the transform of the n-st derivative of a function,

(8.3)

In the case that the terms in the summatory are zero the relation is particularly simple, and for those kind of functions the derivative can be generalized so that this property holds true for non-integer values of a

(8.4)

for which the generalized derivative can be defined as

(8.5)

Keeping in mind the result of the generalized derivative of the exponential (2.1), the following development provides an easy understanding of the reasons involved in the above generalization,

(8.6)

On the other hand, the Fourier transform is defined by

(8.7)

while its inverse transform is

(8.8)

This transform also has an analogous property related to the transform of the n-st derivative of a function,

(8.9)

and the derivative can be generalized so that this property holds true for non-integer values of a

(8.10)

yielding the following definition of the generalized derivative

(8.11)

The application of (2.1) can provide again an easy understanding of the reasons involved in the generalization,

(8.12)

In these two generalizations the implicit limits of differentiation should be determined. In the case of Laplace transform, the generalized derivative is a Riemann-Liouville derivative with a lower limit of 0, whereas in the case of Fourier transform it is a Weyl derivative. Indeed, if we check for instance the derivative generalized by the Fourier transform in the cases of the sine and cosine functions -calculated in (2.4), (2.5), (2.8) and (2.9) with the generalized derivative of the exponential that we have seen in (7.7) that also is a Weyl derivative- we will find that they match perfectly.